In his proof of Szemerédi’s Theorem, Gowers introduced certain norms that are defined on a parallelepiped structure. A natural question is on which sets a parallelepiped structure (and thus a Gowers norm) can be defined. We focus on dimensions $2$ and $3$ and show when this possible, and describe a correspondence between the parallelepiped structures and nilpotent groups.

Let $S$ be an abelian semigroup, and $A$ a finite subset of $S$. The sumset $hA$ consists of all sums of $h$ elements of $A$, with repetitions allowed. Let $\left|hA\right|$ denote the cardinality of $hA$. Elementary lattice point arguments are used to prove that an arbitrary abelian semigroup has polynomial growth, that is, there exists a polynomial $p\left(t\right)$ such that $\left|hA\right|=p\left(h\right)$ for all sufficiently large $h$. Lattice point counting is also used to prove that sumsets of the form $h_1{A}_1+\cdots +h_r{A}_r$ have multivariate polynomial growth.

We extend Hoggar's theorem that the sum of two independent discrete-valued log-concave random variables is itself log-concave. We introduce conditions under which the result still holds for dependent variables. We argue that these conditions are natural by giving some applications. Firstly, we use our main theorem to give simple proofs of the log-concavity of the Stirling numbers of the second kind and of the Eulerian numbers. Secondly, we prove results concerning the log-concavity of the sum of...

We give simple randomized algorithms leading to new upper bounds for combinatorial problems of Choi and Erdős: For an arbitrary additive group G let $P_n\left(G\right)$ denote the set of all subsets S of G with n elements having the property that 0 is not in S+S. Call a subset A of G admissible with respect to a set S from $P_n\left(G\right)$ if the sum of each pair of distinct elements of A lies outside S. Suppose first that S is a subset of the positive integers in the interval [2n,4n). Denote by f(S) the number of elements in a...

Let $\varphi (x_1,...,x_h,y)=u_1{x}_1+\cdots +u_h{x}_h+vy$ be a linear form with nonzero integer coefficients $u_1,...,u_h,v.$ Let $𝒜=(A_1,...,A_h)$ be an $h$-tuple of finite sets of integers and let $B$ be an infinite set of integers. Define the representation function associated to the form $\varphi$ and the sets $𝒜$ and $B$ as follows :$$R_{𝒜,B}^{\left(\varphi \right)}\left(n\right)=\text{card}\left(\begin{array}{cc}\hfill \{(a_1,...,a_h,b)\in A_1& \times \cdots \times A_h\times B:\hfill \\ & \varphi (a_1,...,a_h,b)=n\}\hfill \end{array}\right).$$If this representation function is constant, then the set $B$ is periodic and the period of $B$ will be bounded in terms of the diameter of the finite set $\{\varphi (a_1,...,a_h,0):(a_1,...,a_h)\in A_1\times \cdots \times A_h\}.$ Other results for complementing sets with respect to linear forms are also proved.

We first note that a result of Gowers on product-free sets in groups has an unexpected consequence: If $k$ is the minimal degree of a representation of the finite group $G$, then for every subset $B$ of $G$ with $\left|B\right|>\left|G\right|/k^{1/3}$ we have $B^3=G$. We use this to obtain improved versions of recent deep theorems of Helfgott and of Shalev concerning product decompositions of finite simple groups, with much simpler proofs. On the other hand, we prove a version of Jordan’s theorem which implies that if $k\ge 2$, then $G$ has a proper subgroup...